PS19
ガスジェット浮遊中の溶融金属液滴の表面振動特性の数値解析法
Numerical analysis method for oscillation behavior of
aerodynamically levitated molten metal droplets
小池 聡一郎 1, 小畠 秀和 2, 清宮 優作 3, 小澤 俊平 3, 杉岡 健一 4、石川 毅彦 5、白鳥 英 1
Soichiro KOIKE1, Hidekazu KOBATAKE2, Yusaku SEIMIYA3, Shumpei OZAWA3,Ken-ichi SUGIOKA4, Takehiko ISHIKAWA5, and Suguru SHIRATORI1
東京都市大学，Tokyo City University
同志社大学，Doshisha University
3 千葉工業大学，Chiba Institute of Technology
4 富山県立大学，Toyama Prefectural University
5 JAXA，Institute of Space and Astronautical Science, JAXA12

1. Introduction
For sustainable manned activities on the moon, manufacturing with lunar regolith is essential. Especially,
the melting and solidification process is necessary for the exterior walls so that they can survive in the severe
lunar environment. For this purpose, preliminary numerical simulations play significant roles in the design ofefficient manufacturing processes. Although the main components of the lunar regolith are metal oxides, thethermophysical properties of molten metal oxides have been measured for only a few materials. Therefore,
the measurements of the thermophysical properties of molten metal oxides are an important task. For suchmeasurements, aerodynamic levitation (ADL) is regarded as a suitable method, but there is an uncertainty
problem. Figure 1(a) shows the appearance of droplet during ADL.
In ADL, the oscillatory behavior of a droplet is observed with a camera, and surface tension and viscosity
are calculated based on mathematical models using the obtained frequency and damping time constant. The
relationship between frequency and surface tension was analytically derived by Rayleigh 1), and therelationship between the damping time constant and viscosity was derived by Lamb 2). In these mathematicalmodels, it is assumed that the droplet is a sphere, and that the system is in a stationary state on a time-averaged
basis. In ADL, internal convection occurs due to the Marangoni effect from heating the droplet and the shear
force from the airflow. This convection is thought to alter the frequency and damping time constant, which isbelieved to be the cause of the discrepancies. This study is aimed to establish the relationship between physicalproperties (surface tension and viscosity) and oscillation behavior (frequency and decay time constant).

Fig. 1: (a) Schematics of aero-dynamic levitation. (b) Problems to be considered
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2. Approach
Basic concept
On the surface of a droplet levitated by a gas jet, aerodynamic forces from the airflow and thermal flow
due to laser heating occur. The pressure on the normal droplet surface is determined by the balance between
Laplace pressure due to curvature distribution and hydrostatic pressure, to which pressure from the gas jet isadded. The shear forces acting on the liquid surface include not only the shear force from the airflow but also
the Marangoni effect caused by the surface temperature distribution formed by laser heating. These shear
forces generate internal flow. Constructing a model that considers all these factors is complex; therefore, thisstudy ignores the static deformation of the droplet. Although the deformation of the droplet would
dynamically change the gas jet pressure and shear forces, these are also disregarded for simplicity.
The overall structure of the calculation model, which shown in Figure 1(b), involves decomposing theflow field into a basic flow, assuming an axisymmetric steady state, and fluctuation components from the basic
flow. The basic flow is calculated as a nonlinear problem. The fluctuation components, being assumed small,are linearized, and an eigenvalue problem concerning the fluctuation field is solved to determine theoscillation frequencies and damping time constants.
Mathematical model of surface vibration
The droplet is assumed to be axisymmetric with respect to the direction of gravity. The interface positionof the droplet is implicit function representation as shown in equation (1).
𝐹(𝑟, 𝜃, 𝜑, 𝑡) = 𝑟 − 𝑅0 (1 + 𝑟 ′ (𝜃, 𝜑, 𝑡)),(1)

where 𝑅0 is the radius of the corresponding sphere, and 𝑟’ is the relative deformation from the perfect sphere.
The flow field is divided into a basic flow and oscillatory components and is expressed in spherical coordinates
as shown in equation (2).
𝒖(𝑟, 𝜃, 𝜑, 𝑡) = 𝒖𝟎 (𝑟, 𝜃) + 𝒖′ (𝑟, 𝜃, 𝜑, 𝑡){ 𝒑(𝑟, 𝜃, 𝜑, 𝑡) = 𝒑𝟎 (𝑟, 𝜃) + 𝒑′ (𝑟, 𝜃, 𝜑, 𝑡) ,𝚯(𝑟, 𝜃, 𝜑, 𝑡) = 𝚯𝟎 (𝑟, 𝜃) + 𝚯′ (𝑟, 𝜃, 𝜑, 𝑡)(𝟐)

For spatial discretization, the spectral method is used, expanding as shown in equation (3).
𝑁𝑟 𝑁𝑙𝑙𝑢(𝑟, 𝜃, 𝜑) = ∑ ∑ ∑ 𝑢𝑖𝑙𝑚 (𝑡)𝑇𝑖 (𝑟)𝑌𝑙𝑚 (𝜃, 𝜑) ,𝑖(3)𝑙=𝑜 𝑚=−𝑙In the 𝑟 direction, Chebyshev polynomials are used, and in the 𝜃 and 𝜙 directions, spherical harmonics areused as basis functions. For the discretized equations, the basic flow is computed iteratively using the NewtonRaphson method. Assuming the following normal modes in equation (4) for the linearized perturbation
equations, they are solved as a generalized complex eigenvalue problem.

Fig.2: Surface oscillation and disturbance velocity fields.
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Fig.3: (a,b) Relative errors in oscillation frequencies from Rayleigh’s frequency. (c,d) Relative errors intime constant for decay from Lamb’s theory.

𝒖′ (𝑟, 𝜃, 𝜑, 𝑡).

𝒖
̂ (𝑟, 𝜃)( 𝑝′ (𝑟, 𝜃, 𝜑, 𝑡) ) = ( 𝑝̂ (𝑟, 𝜃) ) exp(𝜆𝑡 + 𝑖𝑚𝜑) + c. c. ,̂ (𝑟, 𝜃)Θ
Θ′ (𝑟, 𝜃, 𝜑, 𝑡)(4)Solving the eigenvalue problem yields the decay time constant 𝜏 and the frequency 𝜔.
3. ResultsFig.3: (a,b) Relative errors in oscillation frequencies from Rayleigh’s frequency. (c,d) Relative errors intime constant for decay from Lamb’s theory. We performed numerical simulations assuming a gas jet
levitation experiment of molten Ni. Fig.2 shows the dynamic deformation of the droplet obtained as
eigenvectors, with pressure (color) and velocity (vectors) of the disturbance field. Fig.3 shows the oscillation
frequency (a, b) and decay time constant (c, d) when varying the range of the Marangoni effect and the strengthof airflow shear, displayed as the relative error from the Rayleigh oscillation frequency and Lamb's decay timeconstant. For both the Marangoni effect and airflow shear, there was little change observed in the vibration
frequency. However, the damping time constant showed different trends depending on the azimuthal
wavenumber m. The 𝑚 = 0 and 𝑚 = 1 modes exhibited a decreasing trend, while the 𝑚 = 2 mode showed
an increasing trend. When combining all modes, the overall damping tended to accelerate, which isqualitatively consistent with the trends observed on the ISS (3).
Acknowledgments
This study was partly supported by the Grant-in-Aid for Front Loading Research from the Advisory
Committee for Space Utilization Research in ISAS/JAXA.
References
1)
2)
3)

L. Rayleigh: Phil. Mag. 32, pp. 529–546 (1916).
H. Lamb: Hydrodynamics，6th ed. Cambridge University Press (1932)
T. Ishikawa et al.: Int. J. Microgravity Sci. Appl. 39(1), p.390101 (2022).
© 2024 by the authors. Submitted for possible open access publication under
the terms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

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